Section 3.5The Area Under the Velocity-Time Graph
3.5The Area Under the Velocity-Time Graph
A natural question to ask about falling objects is how fast they fall, but
Galileo showed that the question has no answer. The physical law that he
discovered connects a cause (the attraction of the planet Earthís mass) to an
effect, but the effect is predicted in terms of an acceleration rather than a
velocity. In fact, no physical law predicts a definite velocity as a result of a
specific phenomenon, because velocity cannot be measured in absolute
terms, and only changes in velocity relate directly to physical phenomena.
The unfortunate thing about this situation is that the definitions of
velocity and acceleration are stated in terms of the tangent-line technique,
which lets you go from x to v to a, but not the other way around. Without a
technique to go backwards from a to v to x, we cannot say anything quanti-
tative, for instance, about the x-t graph of a falling object. Such a technique
does exist, and I used it to make the x-t graphs in all the examples above.
First letís concentrate on how to get x information out of a v-t graph. In
example (a), an object moves at a speed of 20 m/s for a period of 4.0 s. The
distance covered is
(4.0 s)=80 m. Notice that the quanti-
ties being multiplied are the width and the height of the shaded rectangle
ó or, strictly speaking, the time represented by its width and the velocity
represented by its height. The distance of
x=80 m thus corresponds to the
area of the shaded part of the graph.
The next step in sophistication is an example like (b), where the object
moves at a constant speed of 10 m/s for two seconds, then for two seconds
at a different constant speed of 20 m/s. The shaded region can be split into
a small rectangle on the left, with an area representing
x=20 m, and a taller
one on the right, corresponding to another 40 m of motion. The total
distance is thus 60 m, which corresponds to the total area under the graph.
An example like (c) is now just a trivial generalization; there is simply a
large number of skinny rectangular areas to add up. But notice that graph
(c) is quite a good approximation to the smooth curve (d). Even though we
have no formula for the area of a funny shape like (d), we can approximate
its area by dividing it up into smaller areas like rectangles, whose area is
easier to calculate. If someone hands you a graph like (d) and asks you to
find the area under it, the simplest approach is just to count up the little
rectangles on the underlying graph paper, making rough estimates of
fractional rectangles as you go along.